Question 22.3: Consider a 2 × 2 multivariable system, having state space mo...
Consider a 2 × 2 multivariable system, having state space model given by
A_{o}=\begin{bmatrix}1 & 1 & 1 \\ 2 & -1 & 0 \\ 3 & -2 & 2 \end{bmatrix}; B_{o}=\begin{bmatrix} 0 & 1 \\ 1 & 0 \\ 2 & -1\end{bmatrix}; C_{o}=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}; D_{o}=0Find a state feedback gain matrix K such that the closed loop poles are all located in the disk with center at (−α; 0) and radius ρ,where α = 6 and ρ = 2.
We use the approach proposed above, i.e. we transform the complex variable s according to (22.8.4) and then solve a discrete time optimal regulator problem.
We first need the state space representation in the transformed space. This is evaluated by applying equation (22.8.5), which leads to
\zeta =\frac{s+\alpha }{\rho } (22.8.4)
\zeta X(\zeta )=\frac{1}{\rho }(\alpha I+A_{o})X(\zeta )+\frac{1}{\rho }B_{o}U(\zeta ) (22.8.5)
A_{\zeta }=\frac{1}{\rho }(\alpha I+A_{o}) and B_{\zeta }=\frac{1}{\rho }B_{o} (22.8.7)
The MATLAB command dlqr, with weighting matrices \Psi =I_{3} and \Phi =I_{2}, is then used to obtain the optimal gain K_{\zeta }, which is
K_{\zeta }=\begin{bmatrix} 7.00 & -4.58 & 7.73 \\ 3.18 & 7.02 & -4.10 \end{bmatrix} (22.8.8)
When this optimal gain is used in the original continuous time system, the closed loop poles, computed from det(sI-A_{o}+B_{o}K_{\zeta })=0, are located at −5.13, −5.45 and −5.59. All these poles lie in the prescribed region, as expected.